Anomalous transport in a model of Hamiltonian round-off

Abstract

We study the propagation of round-off error near the periodic orbits of a linear area-preserving map - a planar rotation by a rational angle - which is discretized on a lattice in such a way as to retain invertibility. We consider the round-off error probability distribution as a function of time t, and we show that for each t this is an algebraic number, which can be calculated exactly. We prove that its kth moment increases asymptotically as , where is the fractional dimension of a self-similar set related to periodic orbits of long-period, while G is a bounded function, periodic in the logarithm of t. This implies the diffusion coefficient displays bounded variations, while all higher order transport coefficients diverge, resulting in anomalous transport. This result contrasts with the case of irrational rotations, where the existence of a central limit theorem has been recently established (Vladimirov I 1996 Preprint Deakin University).